Covariance Structure Maximum-Likelihood Estimates in Compound Gaussian Noise: Existence and Algorithm Analysis

International audienceRecently, a new adaptive scheme [1], [2] has been introduced for covariance structure matrix estimation in the context of adaptive radar detection under non Gaussian noise. This latter has been modelled by compound-Gaussian noise, which is the product c of the square root of a positive unknown variable τ (deterministic or random) and an independent Gaussian vector x, c = √ τ x. Because of the implicit algebraic structure of the equation to solve, we called the corresponding solution, the Fixed Point (FP) estimate. When τ is assumed deterministic and unknown, the FP is the exact Maximum Likelihood (ML) estimate of the noise covariance structure, while when τ is a positive random variable, the FP is an Approximate Maximum Likelihood (AML). This estimate has been already used for its excellent statistical properties without proofs of its existence and uniqueness. The major contribution of this paper is to fill these gaps. Our derivation is based on some Likelihood functions general properties like homogeneity and can be easily adapted to other recursive contexts. Moreover, the corresponding iterative algorithm used for the FP estimate practical determination is also analyzed and we show the convergence of this recursive scheme, ensured whatever the initialization